Editing Flat module (section)

=== Examples ===
* A ring homomorphism <math>R\to S</math> such that <math>S</math> is a nonzero free {{mvar|R}}-module is faithfully flat. For example:
** Every [[field extension]] is faithfully flat. This property is implicitly behind the use of [[complexification]] for proving results on real vector spaces.
** A [[polynomial ring]] is a faithfully flat extension of its ring of coefficients.
** If <math>p\in R[x]</math> is a [[monic polynomial]], the inclusion <math>R \hookrightarrow R[t]/\langle p \rangle</math> is faithfully flat.
* Let <math>t_1, \ldots, t_k\in R.</math> The [[direct product]] <math>\textstyle\prod_i R[t_i^{-1}]</math> of the [[localization (commutative algebra)|localizations]] at the <math>t_i</math> is faithfully flat over <math>R</math> if and only if <math>t_1, \ldots, t_k</math> generate the [[unit ideal]] of <math>R</math> (that is, if <math>1</math> is a [[linear combination]] of the <math>t_i</math>).{{sfn|Artin|1999|loc=Exercise (3) after Proposition III.5.2|ps=none}}
* The [[direct sum]] of the localizations <math>R_\mathfrak p</math> of <math>R</math> at all its prime ideals is a faithfully flat module that is not an algebra, except if there are finitely many prime ideals.

The two last examples are implicitly behind the wide use of localization in commutative algebra and algebraic geometry.

* For a given ring homomorphism <math>f: A \to B,</math> there is an associated complex called the [[Amitsur complex]]:<ref>{{cite web |url=https://ncatlab.org/nlab/show/Amitsur+complex |title=Amitsur Complex |website=ncatlab.org}}</ref>
<math display="block">0 \to A \overset{f}\to B \overset{\delta^0}\to B \otimes_A B \overset{\delta^1}\to B \otimes_A B \otimes_A B \to \cdots</math>
where the coboundary operators <math>\delta^n</math> are the alternating sums of the maps obtained by inserting 1 in each spot; e.g., <math>\delta^0(b) = b \otimes 1-1 \otimes b</math>. Then (Grothendieck) this complex is exact if <math>f</math> is faithfully flat.